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G = C3×Dic10order 120 = 23·3·5

Direct product of C3 and Dic10

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×Dic10, C153Q8, C60.3C2, C20.1C6, C12.3D5, C6.13D10, Dic5.1C6, C30.13C22, C5⋊(C3×Q8), C4.(C3×D5), C2.3(C6×D5), C10.1(C2×C6), (C3×Dic5).2C2, SmallGroup(120,16)

Series: Derived Chief Lower central Upper central

C1C10 — C3×Dic10
C1C5C10C30C3×Dic5 — C3×Dic10
C5C10 — C3×Dic10
C1C6C12

Generators and relations for C3×Dic10
 G = < a,b,c | a3=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C4
5Q8
5C12
5C12
5C3×Q8

Smallest permutation representation of C3×Dic10
Regular action on 120 points
Generators in S120
(1 84 66)(2 85 67)(3 86 68)(4 87 69)(5 88 70)(6 89 71)(7 90 72)(8 91 73)(9 92 74)(10 93 75)(11 94 76)(12 95 77)(13 96 78)(14 97 79)(15 98 80)(16 99 61)(17 100 62)(18 81 63)(19 82 64)(20 83 65)(21 42 107)(22 43 108)(23 44 109)(24 45 110)(25 46 111)(26 47 112)(27 48 113)(28 49 114)(29 50 115)(30 51 116)(31 52 117)(32 53 118)(33 54 119)(34 55 120)(35 56 101)(36 57 102)(37 58 103)(38 59 104)(39 60 105)(40 41 106)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 44 11 54)(2 43 12 53)(3 42 13 52)(4 41 14 51)(5 60 15 50)(6 59 16 49)(7 58 17 48)(8 57 18 47)(9 56 19 46)(10 55 20 45)(21 78 31 68)(22 77 32 67)(23 76 33 66)(24 75 34 65)(25 74 35 64)(26 73 36 63)(27 72 37 62)(28 71 38 61)(29 70 39 80)(30 69 40 79)(81 112 91 102)(82 111 92 101)(83 110 93 120)(84 109 94 119)(85 108 95 118)(86 107 96 117)(87 106 97 116)(88 105 98 115)(89 104 99 114)(90 103 100 113)

G:=sub<Sym(120)| (1,84,66)(2,85,67)(3,86,68)(4,87,69)(5,88,70)(6,89,71)(7,90,72)(8,91,73)(9,92,74)(10,93,75)(11,94,76)(12,95,77)(13,96,78)(14,97,79)(15,98,80)(16,99,61)(17,100,62)(18,81,63)(19,82,64)(20,83,65)(21,42,107)(22,43,108)(23,44,109)(24,45,110)(25,46,111)(26,47,112)(27,48,113)(28,49,114)(29,50,115)(30,51,116)(31,52,117)(32,53,118)(33,54,119)(34,55,120)(35,56,101)(36,57,102)(37,58,103)(38,59,104)(39,60,105)(40,41,106), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,44,11,54)(2,43,12,53)(3,42,13,52)(4,41,14,51)(5,60,15,50)(6,59,16,49)(7,58,17,48)(8,57,18,47)(9,56,19,46)(10,55,20,45)(21,78,31,68)(22,77,32,67)(23,76,33,66)(24,75,34,65)(25,74,35,64)(26,73,36,63)(27,72,37,62)(28,71,38,61)(29,70,39,80)(30,69,40,79)(81,112,91,102)(82,111,92,101)(83,110,93,120)(84,109,94,119)(85,108,95,118)(86,107,96,117)(87,106,97,116)(88,105,98,115)(89,104,99,114)(90,103,100,113)>;

G:=Group( (1,84,66)(2,85,67)(3,86,68)(4,87,69)(5,88,70)(6,89,71)(7,90,72)(8,91,73)(9,92,74)(10,93,75)(11,94,76)(12,95,77)(13,96,78)(14,97,79)(15,98,80)(16,99,61)(17,100,62)(18,81,63)(19,82,64)(20,83,65)(21,42,107)(22,43,108)(23,44,109)(24,45,110)(25,46,111)(26,47,112)(27,48,113)(28,49,114)(29,50,115)(30,51,116)(31,52,117)(32,53,118)(33,54,119)(34,55,120)(35,56,101)(36,57,102)(37,58,103)(38,59,104)(39,60,105)(40,41,106), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,44,11,54)(2,43,12,53)(3,42,13,52)(4,41,14,51)(5,60,15,50)(6,59,16,49)(7,58,17,48)(8,57,18,47)(9,56,19,46)(10,55,20,45)(21,78,31,68)(22,77,32,67)(23,76,33,66)(24,75,34,65)(25,74,35,64)(26,73,36,63)(27,72,37,62)(28,71,38,61)(29,70,39,80)(30,69,40,79)(81,112,91,102)(82,111,92,101)(83,110,93,120)(84,109,94,119)(85,108,95,118)(86,107,96,117)(87,106,97,116)(88,105,98,115)(89,104,99,114)(90,103,100,113) );

G=PermutationGroup([(1,84,66),(2,85,67),(3,86,68),(4,87,69),(5,88,70),(6,89,71),(7,90,72),(8,91,73),(9,92,74),(10,93,75),(11,94,76),(12,95,77),(13,96,78),(14,97,79),(15,98,80),(16,99,61),(17,100,62),(18,81,63),(19,82,64),(20,83,65),(21,42,107),(22,43,108),(23,44,109),(24,45,110),(25,46,111),(26,47,112),(27,48,113),(28,49,114),(29,50,115),(30,51,116),(31,52,117),(32,53,118),(33,54,119),(34,55,120),(35,56,101),(36,57,102),(37,58,103),(38,59,104),(39,60,105),(40,41,106)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,44,11,54),(2,43,12,53),(3,42,13,52),(4,41,14,51),(5,60,15,50),(6,59,16,49),(7,58,17,48),(8,57,18,47),(9,56,19,46),(10,55,20,45),(21,78,31,68),(22,77,32,67),(23,76,33,66),(24,75,34,65),(25,74,35,64),(26,73,36,63),(27,72,37,62),(28,71,38,61),(29,70,39,80),(30,69,40,79),(81,112,91,102),(82,111,92,101),(83,110,93,120),(84,109,94,119),(85,108,95,118),(86,107,96,117),(87,106,97,116),(88,105,98,115),(89,104,99,114),(90,103,100,113)])

C3×Dic10 is a maximal subgroup of
C20.D6  C15⋊SD16  C15⋊Q16  C3⋊Dic20  D12⋊D5  D60⋊C2  D15⋊Q8  C3×Q8×D5  Dic10.A4

39 conjugacy classes

class 1  2 3A3B4A4B4C5A5B6A6B10A10B12A12B12C12D12E12F15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order12334445566101012121212121215151515202020203030303060···60
size11112101022112222101010102222222222222···2

39 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C3C6C6Q8D5D10C3×Q8C3×D5Dic10C6×D5C3×Dic10
kernelC3×Dic10C3×Dic5C60Dic10Dic5C20C15C12C6C5C4C3C2C1
# reps12124212224448

Matrix representation of C3×Dic10 in GL2(𝔽19) generated by

70
07
,
811
120
,
114
178
G:=sub<GL(2,GF(19))| [7,0,0,7],[8,12,11,0],[11,17,4,8] >;

C3×Dic10 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{10}
% in TeX

G:=Group("C3xDic10");
// GroupNames label

G:=SmallGroup(120,16);
// by ID

G=gap.SmallGroup(120,16);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-5,60,141,66,2404]);
// Polycyclic

G:=Group<a,b,c|a^3=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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Subgroup lattice of C3×Dic10 in TeX

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